Lumped parameter model for the cardiovascular system

The parameters related to the four heart chambers are the passive and active elastances ${\displaystyle EA_{XX}}$ and ${\displaystyle EB_{XX}}$ (where the subscripts vary among ${\displaystyle RA,RV,LA}$ and ${\displaystyle LV}$ if the elastances refer to the right atrium or ventricle or the left atrium or ventricle, respectively) and the unloaded volumes ${\displaystyle V0_{XX}}$. The dynamics of the heart chambers are described by the time-dependent elastance:[5]

${\displaystyle E_{XX}(t)=EB_{XX}+EA_{XX}f_{XX}(t)}$

where ${\displaystyle f_{XX}(t)}$ is a periodic (with period of an heartbeat) time dependent function ranging from ${\displaystyle 0}$ to ${\displaystyle 1}$ that accounts for the activation phases of the heart during a heartbeat. From the above equation, the passive elastance represents the minimum elastance of the heart chamber, whereas the sum of ${\displaystyle EA_{XX}}$ and ${\displaystyle EB_{XX}}$ the maximum elastance of it. The time-dependent elastance allows the computation of the pressure inside a specific heart chamber as follows:[5]

${\displaystyle p_{XX}(t)=E_{XX}(t)(V_{XX}(t)-V0_{XX})}$

where ${\displaystyle V_{XX}(t)}$ is the volume of blood contained in the heart chamber and the volumes for each chamber are the solutions to the following ordinary differential equations that account for inward and outward blood fluxes associated with the heart chamber:[5]

${\displaystyle {\frac {dV_{LA}(t)}{dt}}=Q_{VEN}^{PUL}(t)-Q_{MV}(t)}$

${\displaystyle {\frac {dV_{LV}(t)}{dt}}=Q_{MV}(t)-Q_{AV}(t)}$

${\displaystyle {\frac {dV_{RA}(t)}{dt}}=Q_{VEN}^{SYS}(t)-Q_{TV}(t)}$

${\displaystyle {\frac {dV_{RV}(t)}{dt}}=Q_{TV}(t)-Q_{PV}(t)}$

where ${\displaystyle Q_{MV}(t),Q_{AV}(t),Q_{TV}(t)}$ and ${\displaystyle Q_{PV}(t)}$ are the fluxes through the mitral, aortic, tricuspid and pulmonary valves respectively and ${\displaystyle Q_{VEN}^{PUL}(t)}$ and ${\displaystyle Q_{VEN}^{SYS}(t)}$ are the fluxes through the pulmonary and systemic veins, respectively.[5]

The valves are modeled as diodes and the blood fluxes across the valves depend on the pressure jumps between the upstream and downstream compartment:[5]

${\displaystyle Q_{MV}(t)=Q_{valve}(p_{LA}(t)-p_{LV}(t))\qquad Q_{AV}(t)=Q_{valve}(p_{LV}(t)-p_{AR}^{SYS}(t))}$

${\displaystyle Q_{TV}(t)=Q_{valve}(p_{RA}(t)-p_{RV}(t))\qquad Q_{TV}(t)=Q_{valve}(p_{RV}(t)-p_{AR}^{PUL}(t))}$

where the pressure inside each heart chamber is defined in the previous section, ${\displaystyle p_{AR}^{SYS}(t)}$ and ${\displaystyle p_{AR}^{PUL}(t)}$ are the time-dependent pressures inside the systemic and pulmonary artery compartment and ${\displaystyle Q_{valve}(\Delta p)}$ is the flux across the valve depending on the pressure jump:[5]

${\displaystyle Q_{valve}(\Delta p)={\begin{cases}{\frac {\Delta p}{R_{min}}}\qquad &{\text{if }}\Delta p<0\\{\frac {\Delta p}{R_{max}}}\qquad &{\text{if }}\Delta p\geq 0\end{cases}}}$

where ${\displaystyle R_{min}}$ and ${\displaystyle R_{max}}$ are the resistances of the valves when they are open and closed respectively.[5]

Each compartment of the blood vessels is characterized by a combination of resistances, capacitances and inductances. For example, the arterial systemic circulation can be described by three parameters ${\displaystyle R_{AR}^{SYS},C_{AR}^{SYS}}$ and ${\displaystyle L_{AR}^{SYS}}$ that represent the arterial systemic resistance, capacitance and inductance. The ordinary differential equations that describes the systemic arterial circulation are:[5]

${\displaystyle C_{AR}^{SYS}{\frac {dp_{AR}^{SYS}}{dt}}=Q_{AV}(t)-Q_{AR}^{SYS}(t)}$

${\displaystyle L_{AR}^{SYS}{\frac {dQ_{AR}^{SYS}(t)}{dt}}=-R_{AR}^{SYS}Q_{AR}^{SYS}(t)+p_{AR}^{SYS}(t)-p_{VEN}^{SYS}(t)}$

where ${\displaystyle Q_{AR}^{SYS}(t)}$ is the blood flux across the systemic arterial compartment and ${\displaystyle p_{VEN}^{SYS}(t)}$ is the pressure inside the veins compartment.[5]

Analogous equations with similar notation hold for the other compartments describing the blood circulation.[5]

Assembling the equations described above the following system is obtained: ${\displaystyle \forall \,t\in [0,T]}$ it holds[5]

${\displaystyle {\begin{cases}{\frac {dV_{LA}(t)}{dt}}=Q_{VEN}^{PUL}(t)-Q_{MV}(t)\\{\frac {dV_{LV}(t)}{dt}}=Q_{MV}(t)-Q_{AV}(t)\\C_{AR}^{SYS}{\frac {dp_{AR}^{SYS}(t)}{dt}}=Q_{AV}(t)-Q_{AR}^{SYS}(t)\\L_{AR}^{SYS}{\frac {dQ_{AR}^{SYS}(t)}{dt}}=-R_{AR}^{SYS}Q_{AR}^{SYS}(t)+p_{AR}^{SYS}(t)-p_{VEN}^{SYS}(t)\\C_{VEN}^{SYS}{\frac {dp_{VEN}^{SYS}(t)}{dt}}=Q_{AR}^{SYS}(t)-Q_{VEN}^{SYS}(t)\\L_{VEN}^{SYS}{\frac {dQ_{VEN}^{SYS}(t)}{dt}}=-R_{VEN}^{SYS}Q_{VEN}^{SYS}(t)+p_{VEN}^{SYS}(t)-p_{RA}(t)\\{\frac {dV_{RA}(t)}{dt}}=Q_{VEN}^{SYS}(t)-Q_{TV}(t)\\{\frac {dV_{RV}(t)}{dt}}=Q_{TV}(t)-Q_{PV}(t)\\C_{AR}^{PUL}{\frac {dp_{AR}^{PUL}(t)}{dt}}=Q_{PV}(t)-Q_{AR}^{PUL}(t)\\L_{AR}^{PUL}{\frac {dQ_{AR}^{PUL}(t)}{dt}}=-R_{AR}^{PUL}Q_{AR}^{PUL}(t)+p_{AR}^{PUL}(t)-p_{VEN}^{PUL}(t)\\C_{VEN}^{PUL}{\frac {dp_{VEN}^{PUL}(t)}{dt}}=Q_{AR}^{PUL}(t)-Q_{VEN}^{PUL}(t)\\L_{VEN}^{PUL}{\frac {dQ_{VEN}^{PUL}(t)}{dt}}=-R_{VEN}^{PUL}Q_{VEN}^{PUL}(t)+p_{VEN}^{PUL}(t)-p_{LA}(t)\end{cases}}}$

Outputs of the lumped parameter cardiovascular model: pressures, blood volumes inside the heart chambers and blood fluxes.

with ${\displaystyle T}$ the final time. The first two equations are related to the volumes in the left atrium and ventricles respectively. The equations from the third to the sixth are related to the pressures, and fluxes of the systemic arterial and venous systems. The last equations are related to the right heart and the pulmonary circulation in an analogous way. The system is completed with initial conditions for each of the unknowns.[5]

From a mathematical point of view, the well-posedness of the problem is a consequence of the Cauchy–Lipschitz theorem, so its solution exists and it is unique. The solution of the system is approximated by means of a numerical method. The numerical simulation has to be computed for more than ${\displaystyle 10}$ heartbeats (the final time ${\displaystyle T}$ depends on the number of heartbeats and the heart rate) to approach the limit cycle of the dynamical system, so that the solution behaves in a similar way to a periodic function emulating the periodicity of the cardiac cycle.[5]